Sample Report: Phenotypic Data Analysis Pipeline
Prepared by: Rainfed Breeding Team
24 March, 2021
Load the Required Libraries
> library(easypackages)
> libraries("dplyr", "reshape2", "readxl", "ggpubr","stringr", "ggplot2",
+ "tidyverse","lme4", "data.table", "readr","plotly", "DT",
+ "pheatmap","asreml", "VennDiagram", "patchwork", "heatmaply",
+ "ggcorrplot", "RColorBrewer", "hrbrthemes", "tm", "proustr", "arm")
Online License checked out Wed Mar 24 21:16:18 20211 General Information on Demo Data
Study: Rainfed Rice Trials
Experimental Design: Augmented Randomized Block Design;
4 blocks
1 Replication, 322 entries (un-replicated) and 12 checks (replicated).
Season: Wet-season (WS).
Location: Two NARES Locations in India.
Year: 2019.
Contact Person: Rain-fed Breeding Team
NOTE: Due to IRRI’s data policies, the actual names of lines and complete metadata information is not given in this demo report.
2 Description of Demo Data Set
Demo data set used in this analysis pipeline was evaluated in augmented RCBD experimental design.
The demo data includes data from two environments phenotyped in India for three traits grain yield, plant height and days to flowering.
Besides blocks, information on columns and rows is also given in the data set.
> # Remove previous work
> rm(list=ls())
> # Upload the demo data set
> demo.data<-read.csv(file="~/Documents/GitHub/Analysis-pipeline/Data/demo.data.csv",
+ header = TRUE)
> # Convert variables into appropriate data types
> demo.data$Genotype<-as.factor(demo.data$Genotype) # Genotypes as factor
> demo.data$Block<-as.factor(demo.data$Block) # Block as factor
> demo.data$Row<-as.factor(demo.data$Row) # Row as factor
> demo.data$Column<-as.factor(demo.data$Column) # Column as factor
> # View as data table
> print_table <- function(table, ...){
+ datatable(table, extensions = 'Buttons',
+ options = list(scrollX = TRUE,
+ dom = '<<t>Bp>',
+ buttons = c('copy', 'excel', 'pdf', 'print')), ...)
+ }
> print_table(demo.data[, c(1, 5,6,8,9,10, 11, 12)], editable = 'cell',
+ rownames = FALSE, caption = htmltools::tags$caption("Table: Raw data for grain yield (kg/ha), days to flowering and plant height in two environmnets.",style="color:black; font-size:130%"), filter = 'top')3 Pre-processing of Data: Checking Quality of Phenotypic Data
Here in this step data will be pre-processed and quality of data will be checked, and only quality phenotypes will be advanced for downstream analysis to have more reliable and accurate estimates or predictors.
The steps in pre-processing involves:
- Looking for missing data
- Descriptive statistics for each variable
- Heat-maps of field experimental design
- Data visualisations as box-plots, histograms and QQ plots
- Look and filter for outliers.
3.1 Visualize the Missing Data.
- Here we will check whether the data has any missing values
> # Missing data count across all columns
> demo.data[demo.data==0]<-NA # Converting any values with Zero into NA
> na_count <-data.frame(missing.count=sapply(demo.data, function(y) sum(length(which(is.na(y))))))
> # colSums(is.na(demo.data)) # alternative
> na_count$Variables<-row.names(na_count)
> # Visualize missing data as bar plot
> ggbarplot(na_count, x = "Variables", y = "missing.count",
+ fill="lightblue",
+ color = "lightblue", # Set bar border colors to white
+ x.text.angle = 45 # Rotate vertically x axis texts
+ )+
+ labs(title="Missing Data Points for all Variables",x="Variables", y = "Count")+
+ theme (plot.title = element_text(color="black", size=12,hjust=0.5, face="bold"), # add and modify the title to plot
+ axis.title.x = element_text(color="black", size=12), # add and modify title to x axis
+ axis.title.y = element_text(color="black", size=12))
> # Let us see which one is missing for Plant Height
> #demo.data$Height[which(is.na(demo.data$Height))]
> # let us see the details on this
> # demo.data[216, ]Note: No missing data observed for each varaiable.
3.2 Descriptive Statistics
- Here data is summarized as mean, median, minimum and maximum values, standard deviation (SD), coefficient of variation (CV) and standard error (SE) for grain yield (kg/ha), days to flowering, and plant height (cm).
> # Summary for grain yield
> summary.Yield<-data.frame(demo.data %>%
+ group_by(Environment)%>%
+ summarize(Mean = mean(Yield, na.rm=TRUE),
+ Median= median(Yield, na.rm=TRUE),
+ SD =sd(Yield, na.rm=TRUE),
+ Min.=min(Yield, na.rm=TRUE),
+ Max.=max(Yield, na.rm=TRUE),
+ CV=sd(Yield, na.rm=TRUE)/mean(Yield, na.rm=TRUE)*100,
+ St.err= sd(Yield, na.rm=TRUE)/sqrt(length(Yield))
+ ))
> summary.Yield<-data.frame(lapply(summary.Yield, function(y) if(is.numeric(y)) round(y, 2) else y))
>
> summary.Yield<-cbind(data.frame(Trait=c(rep("Yield", nrow(summary.Yield)))),summary.Yield )
> # Summary for flowering date
> summary.flowering<-data.frame(demo.data %>%
+ group_by(Environment)%>%
+ summarize(Mean = mean(Days.to.flowering, na.rm=TRUE),
+ Median= median(Days.to.flowering, na.rm=TRUE),
+ SD =sd(Days.to.flowering, na.rm=TRUE),
+ Min.=min(Days.to.flowering, na.rm=TRUE),
+ Max.=max(Days.to.flowering, na.rm=TRUE),
+ CV=sd(Days.to.flowering, na.rm=TRUE)/mean(Days.to.flowering, na.rm=TRUE)*100,
+ St.err= sd(Days.to.flowering, na.rm=TRUE)/sqrt(length(Days.to.flowering))
+ ))
> summary.flowering<-data.frame(lapply(summary.flowering, function(y) if(is.numeric(y)) round(y, 2) else y))
> summary.flowering<-cbind(data.frame(Trait=c(rep("Flowering", nrow(summary.flowering)))),summary.flowering )
> # Summary for plant height
> summary.height<-data.frame(demo.data %>%
+ group_by(Environment)%>%
+ summarize(Mean = mean(Height, na.rm=TRUE),
+ Median= median(Height, na.rm=TRUE),
+ SD =sd(Height, na.rm=TRUE),
+ Min.=min(Height, na.rm=TRUE),
+ Max.=max(Height, na.rm=TRUE),
+ CV=sd(Height, na.rm=TRUE)/mean(Height, na.rm=TRUE)*100,
+ St.err= sd(Height, na.rm=TRUE)/sqrt(length(Height))
+ ))
> summary.height<-cbind(data.frame(Trait=c(rep("Height", nrow(summary.height)))),summary.height )
> # Now combine the all data summeries and view as table
> summary.data<-rbind(summary.Yield, summary.flowering, summary.height)
> summary.data<-data.frame(lapply(summary.data, function(y) if(is.numeric(y)) round(y, 2) else y))
> # Add options to print and export
> print_table(summary.data, rownames = FALSE,caption = htmltools::tags$caption("Data summary including mean, median, standard deviation (SD), coefficient of variation (CV), and standard error (St.err) for yield (kg/ha), days to flowering and plant height (cm).", style="color:black; font-size:130%"))Note: High CV for grain yield is observed as compared to flowering and plant height. Across the two environments CV is differes slightly
3.3 Heat Maps of the Field Experimental Design
- Experimental design in the field for grain yield is visualized through heat map to get better idea about the field design and spatial variations in the field.
3.3.1 Showing heat map of field design under environment 1
>
> # Under environment 1
> demo.data.env1<- subset(demo.data, Environment=="Env1",
+ select =c("Block", "Column", "Yield") )
> demo.data.env1<-data.frame(demo.data.env1%>% group_by(Block)%>% arrange(Block) %>%arrange(Column))
> demo.data.env1<-droplevels.data.frame(demo.data.env1)
> demo.data.env1<-reshape(demo.data.env1, idvar = "Block",
+ timevar = "Column", direction = "wide")
> row.names(demo.data.env1)<-paste0("Block", demo.data.env1$Block)
> demo.data.env1<-data.matrix(demo.data.env1[,-1])
> colnames(demo.data.env1) <- gsub(x = colnames(demo.data.env1),
+ pattern = "Yield.", replacement = "")
>
> plot.gy.env1<-heatmaply(demo.data.env1, main = "Grain yield under environment 1",
+ xlab = "Columns",
+ ylab = "Rows",
+ Rowv=FALSE,
+ Colv = FALSE, cexRow = 0.8, cexCol = 0.6, na.value="white")
> plot.gy.env1Note: Extreme blue shows low grain yield values and yellow are very high grain yield values. X axis shows the list of columns and y-axis the blocks (blocks here are also treated as rows).
3.3.2 Showing heat map of field design under environment 2
> # For Environment 2
> demo.data.env2<- subset(demo.data, Environment=="Env2",
+ select =c("Block", "Column", "Yield") )
> demo.data.env2<-data.frame(demo.data.env2%>% group_by(Block)%>% arrange(Block) %>%arrange(Column))
> demo.data.env2<-droplevels.data.frame(demo.data.env2)
> demo.data.env2<-reshape(demo.data.env2, idvar = "Block",
+ timevar = "Column", direction = "wide")
> row.names(demo.data.env2)<-paste0("Block",demo.data.env2$Block)
> demo.data.env2<-data.matrix(demo.data.env2[,-1])
> colnames(demo.data.env2) <- gsub(x = colnames(demo.data.env2), pattern = "Yield.", replacement = "")
> plot.gy.env2<-heatmaply(demo.data.env2, main = "Grain yield under environment 2",
+ xlab = "Columns",
+ ylab = "Rows",
+ Rowv=FALSE,
+ Colv = FALSE, cexRow = 0.8, cexCol = 0.6, na.value="white")
> plot.gy.env2Note: Extreme blue shows low grain yield values and yellow are very high grain yield values. X axis shows the list of columns and y-axis the blocks (blocks here are also treated as rows). Note down the high single grain yield value in Block 3.
3.4 Data Visualization
- Here in this section we will visualize the data using Box plot, Histograms, and QQ plot.
3.4.1 Box plot
> # First let us visualize the data using boxplots
> myboxplot<- function(dataframe,x,y){
+ aaa <- enquo(x)
+ bbb <- enquo(y)
+ dfname <- enquo(dataframe)
+ dataframe %>%
+ filter(!is.na(!! aaa), !is.na(!! bbb)) %>%
+ #group_by(!! aaa,!! bbb) %>%
+ #count() %>%
+ ggplot(aes_(fill=aaa, x=aaa, y=bbb))+
+ theme_classic()+
+ geom_boxplot()+
+ theme(axis.text.x = element_text(angle = 45, hjust = 1)) +# fill by timepoint to give different color
+ #scale_fill_manual(values = c("", ""))+
+ #scale_color_manual(values = c("", ""))
+ theme (plot.title = element_text(color="black", size=12,hjust=0.5, face = "bold"), # add and modify the title to plot
+ axis.title.x = element_text(color="black", size=12, face = "bold"), # add and modify title to x axis
+ axis.title.y = element_text(color="black", size=12, face="bold")) + # add and modify title to y axis
+ #scale_y_continuous(limits=c(0,15000), breaks=seq(0,15000,1000), expand = c(0, 0))+
+ theme(axis.text= element_text(color = "black", size = 10))+ # modify the axis text
+ theme(legend.title = element_text(colour="black", size=16), legend.position = "none",
+ legend.text = element_text(colour="black", size=14))+ # add and modify the legends
+ guides(fill=guide_legend(title="Environments"))+
+ stat_summary(fun.y=mean, geom="line", aes(group=1)) +
+ stat_summary(fun=mean, geom="point")
+ }
>
> # Now draw the box plot for yield
> p1<-boxplot.yield<-myboxplot(demo.data,x=Environment,y=Yield)+
+ labs(title="",x="Environments", y = "Grain Yield (kg/ha)")+
+ stat_compare_means(method = "anova", label.x = 1.6, label.y = 10000)
> #p1<-ggplotly(boxplot.yield)
>
> # Now draw the box plot for flowering
> p2<-boxplot.flowering<-myboxplot(demo.data,x=Environment,y=Days.to.flowering)+
+ labs(title="",x="Environments", y = "Days to flowering")+
+ stat_compare_means(method = "anova", label.x = 1.6, label.y = 130)
> #p2<-ggplotly(boxplot.flowering)
>
> # Now draw the box plot height
> p3<-boxplot.height<-myboxplot(demo.data,x=Environment,y=Height)+
+ labs(title="",x="Environments", y = "Plant Height (cm)")+
+ stat_compare_means(method = "anova", label.x = 1.6, label.y = 167)
> #p3<-ggplotly(boxplot.height)
> #p1+p2+p3
> par(mfrow=c(1,3))
> p1<-ggplotly(p1)
> p2<-ggplotly(p2)
> p3<-ggplotly(p3)
> subplot(p1, p2, p3, nrows=1, margin = 0.05, titleY = TRUE)Note: Non-significant difference between two environments for yield and plant height, p-value is provided on top of each plot. Outliers are present as black solid dots for all traits.
Histograms and QQ plots are also available , click the buttons below
3.4.2 Histogram plots
- Histograms for all traits to check distribution of data.
> par(mfrow=c(1,2))
> # For grain yield
> envi<-unique(demo.data$Environment)
> for(i in 1:length(envi)){
+ level_envi <- demo.data[which(demo.data$Environment==envi[i]),]
+ hist(level_envi$Yield, col = "pink", xlab="Grain yield (kg/ha)",
+ main=paste(envi[i]))
+
+ }
Showing histograms for Grain yield, flowering and plant height.
> # For Flowering date
> envi<-unique(demo.data$Environment)
> for(i in 1:length(envi)){
+ level_envi <- demo.data[which(demo.data$Environment==envi[i]),]
+ hist(level_envi$Days.to.flowering, col = "pink", xlab="Days to flowering",
+ main=paste(envi[i]))
+
+ }
Showing histograms for Grain yield, flowering and plant height.
> # For Plant height
> envi<-unique(demo.data$Environment)
> for(i in 1:length(envi)){
+ level_envi <- demo.data[which(demo.data$Environment==envi[i]),]
+ hist(level_envi$Height, col = "pink", xlab="Plant Height (cm)",
+ main=paste(envi[i]))
+
+ }
Showing histograms for Grain yield, flowering and plant height.
Showing histograms for grain yield, flowering and plant height under two environments.
3.4.3 QQ plots
- QQ plots are drawn to check the normality of the data. It is just to to see if our data assumptions are plausible.
- We expect line to be straight, if it deviates then it indicates some issues with data. More information on QQ plots can be found here on this link
> ## QQ plots to check normality assumption
> # For the grain Yield
> par(mfrow=c(1,2))
> envi<-unique(demo.data$Environment)
> for(i in 1:length(envi)){
+ level_envi <- demo.data[which(demo.data$Envi==envi[i]),]
+ qqnorm(level_envi$Yield, pch = 1, frame = TRUE, main=paste(envi[i],".Yield"))
+ qqline(level_envi$Yield, col = "steelblue", lwd = 2)
+ }
> # For the days to flowering
> par(mfrow=c(1,2))
> envi<-unique(demo.data$Environment)
> for(i in 1:length(envi)){
+ level_envi <- demo.data[which(demo.data$Envi==envi[i]),]
+ qqnorm(level_envi$Days.to.flowering, pch = 1, frame = TRUE, main=paste(envi[i],".Flowering"))
+ qqline(level_envi$Days.to.flowering, col = "steelblue", lwd = 2)
+ }
> # For plant height
> par(mfrow=c(1,2))
> envi<-unique(demo.data$Environment)
> for(i in 1:length(envi)){
+ level_envi <- demo.data[which(demo.data$Envi==envi[i]),]
+ qqnorm(level_envi$Height, pch = 1, frame = TRUE, main=paste(envi[i],".Height"))
+ qqline(level_envi$Height, col = "steelblue", lwd = 2)
+ }
Grain yield looks good, check the deviation of observed values for plant height and flowering date.
3.5 Identify and Remove Outliers
Note: Outliers may drastically change the estimates, ranking (BLUPs or BLUEs) and predictions!! Further reading Resource 1; Resource 2; Resource 3
- As the data is in Augmented RCBD with one replication, uni-variate approach will be used to find the potential outliers.
- Briefly, we will flag out the lines that lie outside 1.5*IQR, where IQR, the ‘Inter Quartile Range’ is the difference between 75th and 25th quartiles, these are observation that are outside the whiskers in box plot. For more details on this please check this R blog.
- So, in this section, a function named outlier.box which will first subset the data based on environment and replication, then we will get Quantiles, IQR and maximum and minimum values. Then we will flag out and extract the ones that fall outside the maximum and minimum value in the data.
- More robust approach by integrating all the data across all the trials or locations to find potential outliers by the method called Bonferroni–Holm test is briefly described in this article. The R codes on this is available at Click the link.
- Here we will be using Uni-variate approach as we are dealing with un-replicated data sets.
> # Univariate approach to falg out outliers in augmented un-replicated design
> outlier.box<- function(data, trait, name){
+ #test<-subset(data, Envi==envir )# subsset based on environment and replications
+ #test<-droplevels.data.frame(test) # drop factor levels
+ #var_name <- eval(substitute(var),eval(data))
+ trait_name<- eval(substitute(trait),eval(data)) # evaluate trait name
+ Q3 = quantile(trait_name, 0.75, na.rm = TRUE) # get Q3
+ Q1=quantile(trait_name, 0.25, na.rm = TRUE)
+ IQR=IQR(trait_name, na.rm = TRUE)
+ Maxi<-Q3+1.5*IQR # Maximum Value
+ Mini<-Q1-1.5*IQR # Minimum Value
+ #out_flag_max<-ifelse(trait_name >Maxi , "OUTLIER_Max", ".") # Flag lines with maximum value as OUTLIER_Max
+ #out_flag_min <-ifelse(trait_name < Mini , "OUTLIER_Min", ".")
+ out_flag<-ifelse(trait_name >Maxi | trait_name < Mini , name, ".") # Flag the outliers
+ #out<-cbind(out_flag_max,out_flag_min)
+ out_data<-cbind(data, out_flag) # Combine the original data
+ #outliers<- data[which(out_data$out_flag_max!="." |out_data$out_flag_min!="." ), c(1, 2,4,7,15)] # Extract the ones with extreame values and return only selected columns
+ #outliers<- data[which(out_data$out_flag!="."),] # Extract the ones with extreme values and return only selected columns
+ return( out_data)
+ }
> table(demo.data$Environment)
Env1 Env2
380 380
> # Now subset the data and use above function to identify the outliers
> # Subset two environment 1
> Env1<-subset(demo.data, Environment=="Env1") # drought data
> Env1<-droplevels.data.frame(Env1) # drop factor levels
> # Now subset the environment 2
> Env2<-subset(demo.data, Environment=="Env2")
> Env2<-droplevels.data.frame(Env2) # drop factor levels
> # Now identify the outliers for grain yield
> Env1<-outlier.box(Env1,name="Outlier.GY", trait = Yield) # returns the list that has outliers for drought environment
> Env2<-outlier.box(Env2,name="Outlier.GY", trait = Yield) # returns the list that has outliers for non-stress environment
> # Now identify the outliers for plant height
> Env1<-outlier.box(Env1,name="Outlier.PH", trait = Height) # returns the list that has outliers for drought environment
> Env2<-outlier.box(Env2,name="Outlier.PH", trait = Height) # returns the list that has outliers for non-stress environment
> # Now identify the outliers for days to flowering
> Env1<-outlier.box(Env1,name="Outlier.FL", trait = Days.to.flowering) # returns the list that has outliers for drought environment
> Env2<-outlier.box(Env2,name="Outlier.FL", trait = Days.to.flowering) # returns the list that has outliers for non-stress environment
> # Now merge all the files and save them
> demo.data.out<-rbind(Env1, Env2)
> #Here we will inspect all the outliers and filter the extreme ones.
> #First let us change the names of last two columns
> colnames(demo.data.out)[c(13,14,15)] <- c("out.flag.GY", "out.flag.PH", "out.flag.FL")
> # Visualize as table
> print_table(demo.data.out[, c(1, 4, 10,11,12,13,14,15)], editable = 'cell', rownames = FALSE, caption = htmltools::tags$caption("Table: Showing the list of outliers for grain yield, plant height and flowering date in two environments.",style="color:black; font-size:130%"), filter='top')3.6 Filter outliers
- In this section we will convert the outliers into the missing values for all the three traits and save the file for downstream analysis.
- Users can also convert them into mean values. See the codes given below for more details.
> # For grain yield
> demo.data.out$Yield<- ifelse(demo.data.out$out.flag.GY==".", demo.data.out$Yield, NA)
> # For plant height
> demo.data.out$Height<- ifelse(demo.data.out$out.flag.PH==".", demo.data.out$Height, NA)
> # For plant height
> demo.data.out$Days.to.flowering<- ifelse(demo.data.out$out.flag.FL==".", demo.data.out$Days.to.flowering, NA)
> # We can also conver the outliers into mean values
> #data<-data.frame(matrix())
> #env<- unique(TEST$Envi)
> #for(i in 1:length(env)){
> #data1<-TEST[which(TEST$Envi==env[i]),]
> #data1$Yield <- ifelse(data1$out.all==".", data1$Yield, mean(data1$Yield))
> #return(data1)
> #data2<-rbind(data1, data)
> #}Box Plots after Removing Outliers
> # Now draw the box plot
> p1<-boxplot.yield<-myboxplot(demo.data.out,x=Environment,y=Yield)+
+ labs(title="",x="Environments", y = "Grain Yield (kg/ha)")+
+ stat_compare_means(method = "anova", label.x = 1.6, label.y = 10000)
> #p1<-ggplotly(boxplot.yield)
> # Now draw the box plot for flowering
> p2<-boxplot.flowering<-myboxplot(demo.data.out,x=Environment,y=Days.to.flowering)+
+ labs(title="",x="Environments", y = "Days to flowering")+
+ stat_compare_means(method = "anova", label.x = 1.6, label.y = 130)
> #p2<-ggplotly(boxplot.flowering)
> # Now draw the box plot height
> p3<-boxplot.height<-myboxplot(demo.data.out,x=Environment,y=Height)+
+ labs(title="",x="Environments", y = "Plant Height (cm)")+
+ stat_compare_means(method = "anova", label.x = 1.6, label.y = 167)
> #p3<-ggplotly(boxplot.height)
> par(mfrow=c(1,3))
> p1<-ggplotly(p1)
> p2<-ggplotly(p2)
> p3<-ggplotly(p3)
> subplot(p1, p2, p3, nrows=1, margin = 0.05, titleY = TRUE)Box plot showing distribution for all traits.
Note: Seems much better now.
Descriptive Statistics after Removing Outliers
> summary.Yield<-data.frame(demo.data.out %>%
+ group_by(Environment)%>%
+ summarize(Mean = mean(Yield, na.rm=TRUE),
+ Median= median(Yield, na.rm=TRUE),
+ SD =sd(Yield, na.rm=TRUE),
+ Min.=min(Yield, na.rm=TRUE),
+ Max.=max(Yield, na.rm=TRUE),
+ CV=sd(Yield, na.rm=TRUE)/mean(Yield, na.rm=TRUE)*100,
+ St.err= sd(Yield, na.rm=TRUE)/sqrt(length(Yield))
+ ))
> summary.Yield<-data.frame(lapply(summary.Yield, function(y) if(is.numeric(y)) round(y, 2) else y))
>
> summary.Yield<-cbind(data.frame(Trait=c(rep("Yield", nrow(summary.Yield)))),summary.Yield )
> # Summary for the flowering data
> summary.flowering<-data.frame(demo.data.out %>%
+ group_by(Environment)%>%
+ summarize(Mean = mean(Days.to.flowering, na.rm=TRUE),
+ Median= median(Days.to.flowering, na.rm=TRUE),
+ SD =sd(Days.to.flowering, na.rm=TRUE),
+ Min.=min(Days.to.flowering, na.rm=TRUE),
+ Max.=max(Days.to.flowering, na.rm=TRUE),
+ CV=sd(Days.to.flowering, na.rm=TRUE)/mean(Days.to.flowering, na.rm=TRUE)*100,
+ St.err= sd(Days.to.flowering, na.rm=TRUE)/sqrt(length(Days.to.flowering))
+ ))
> summary.flowering<-data.frame(lapply(summary.flowering, function(y) if(is.numeric(y)) round(y, 2) else y))
> summary.flowering<-cbind(data.frame(Trait=c(rep("Flowering", nrow(summary.flowering)))),summary.flowering )
> # Summary for plant height
> summary.height<-data.frame(demo.data.out %>%
+ group_by(Environment)%>%
+ summarize(Mean = mean(Height, na.rm=TRUE),
+ Median= median(Height, na.rm=TRUE),
+ SD =sd(Height, na.rm=TRUE),
+ Min.=min(Height, na.rm=TRUE),
+ Max.=max(Height, na.rm=TRUE),
+ CV=sd(Height, na.rm=TRUE)/mean(Height, na.rm=TRUE)*100,
+ St.err= sd(Height, na.rm=TRUE)/sqrt(length(Height))
+ ))
> summary.height<-data.frame(lapply(summary.height, function(y) if(is.numeric(y)) round(y, 2) else y))
> summary.height<-cbind(data.frame(Trait=c(rep("Height", nrow(summary.height)))),summary.height )
> # Now combine the all data summeries and view as table
> summary.data<-rbind(summary.Yield, summary.flowering, summary.height)
> datatable(summary.data,options = list(pageLength = 7, dom = 'tip'), rownames = FALSE,caption = htmltools::tags$caption("Data summary after removing outliers.", style="color:black; font-size:130%"))4 Data Analysis for Grain Yield in ASReml R Package
In this section, data analysis will be shown only for grain yield trait using a Linear Mixed-Model Approach in ASReml-R package.
Demo data analysis for grain yield will also be shown using freely available lme4 R Package package, and will be useful to the users which do not have access to the commercial ASReml-R package. See the section 5 below.
In general analysis is divided in two parts: First Separate analysis or step-wise analysis: In this two environments will be analyzed separately. We will be testing five mixed models correcting for experimental design factors (Blocks here) and spatial trends in the field. Then best model will be selected (model having lowest AIC value ) and used to extract the BLUPs or Breeding Values and Heritability. Second Combined analysis or Multi-environment trial (MET) analysis: In this analysis two environments will combined and analyzed together and single value BLUPs for each genotype will be extracted. The model used for combined analysis is again Mixed-Model accounting for spatial trends. We already know best spatial model (found in separate analysis above) in two environments, so this information will be used and incorporated in combined mixed-model analysis model.
Then we will rank the genotypes based on the BLUP values and compare it with checks.
We will also look at correlations between the environments.
4.1 Separate Analysis
As mentioned above only Five models will be used to account for experimental design factors and accounting for spatial variations.
The five models shown here are for demo purpose, more models can be used to model the phenotypic data. For more information on these models and other advanced additional mixed models is available here: Asreml-R-Tutorial: Go to section 4.1; Book: Genetic Data Analysis for Plant and Animal Breeding; Chapter 7.
Click button below for more description on the models:
Model 1
In this model we account for just experimental design factor Block and no spatial effects.
Note we used block as fixed effect in most cases due to less than 5 degrees of freedom. If you are interested to know whether to use block fixed or random in model I highly recommend this Blocks Fixed or Random?
Also Note row and block is same in all the trials. So it does not matter whether we use row or block in model.
Best linear unbiased predictors (BLUPs) extracted here is equivalent to breeding values
\[ Y_{ij}= \mu+G_{i} + B_{j} + \varepsilon_{ij}\\ Y_{ij}= \text{ is the effect of $i$th genotype in $j$th block} \\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ B_{k}= \text {fixed effect of $k$th block}\\ \varepsilon_{ij}=\text{error}\\ \text{here we assume errors are independent and identically distributed }\varepsilon\sim \text{$iid$N}(0,\sigma_e^2)\\ \]
R script in Asreml
model1<-asreml(fixed=trait~Block, random=~Genotype, na.method=“include”, data=data)
- In the above model block is treated as fixed effect and genotype as random effect
Model 2
- In this model we account for just experimental design factor block, and column no spatial effects.
\[ Y_{ijk}= \mu+G_{i} + B_{j}+ C_{k} + \varepsilon_{ijk}\\ Y_{ijk}= \text{ is the effect of $i$th genotype in $j$th block and $k$th column} \\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ B_{j}= \text {random efect of $j$th block}\\ C_{k}= \text {random efect of $k$th column}\\ \varepsilon_{ijk}=\text{error}\\ \text{here we assume residuals are independent and identically distributed }\varepsilon\sim \text{$iid$N}(0,\sigma_e^2)\\ \]
R script in Asreml
model2<-asreml(fixed=trait~1, random=~Column+Block+Genotype, na.method=“include”, data=data)
- Block, column and genotype all are used as random effects.
Model 3
- In this model we account block, and column effects, and for spatial trends/effects i.e, correlated residuals both across rows and columns.
\[ Y_{ijk}= G_{i} + B_{j}+ C_{k} + \xi\\ Y_{ijk}= \text{ is the effect of $i$th genotype in $j$th block and $k$th column} \\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ B_{j}= \text {random efect of $j$th block}\\ C_{k}= \text {random efect of $k$th column}\\ \xi=\text {residual error}\\ \]
here, we assume \(\xi\) is a random effect that represents correlated residuals based on the distance between plots along both the rows and columns, where, \(\xi\sim N(0,\mathbf{R})\) and R is the covariance matrix of \(\xi\). The difference between this model and model 1 and model 2 described above is the structure of the covariance residuals \(R ={\sigma_\xi^2\ \Sigma}_c\left(\rho_c\right)\otimes\Sigma_r\left(\rho_r\right)\). \(\sigma_\xi^2\) is the variance of spatially dependent residual; \({\Sigma}_c\left(\rho_c\right)\) and \(\ \Sigma_r\left(\rho_r\right)\) represents the first-order autoregressive correlation matrices and \(\rho_{c\ }\) and \(\rho_{r\ }\) are the autocorrelation parameters for the columns and rows; \(\otimes\) represents the Kronecker product between separable auto-regressive processes of the first order in the row-column dimensions. For more details on this, these references would be helpful Gilmour et al., 1997; Gogel et al., 2018; Andrade et al., 2020; Bernardeli et al.202
R script in Asreml
model3<-asreml(fixed=trait~1, random=~Column+Block+Genotype,residual =~ar1v(Block):ar1(Column), na.method=“include”, data=data)
- Block, column and genotype all are used as random effects.
Model 4
- In this model we account only for fixed block effect and spatial trends i.e, correlated residuals both across blocks or rows and columns in row and column directions both.
\[ Y_{ijk}= G_{i} + B_{j} + \xi\\ Y_{ijk}= \text{ is the effect of $i$th genotype in $j$th block} \\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ B_{k}= \text {fixed efect of $j$th block}\\ \xi=\text {residual error}\\ \] here, we assume \(\xi\) is a random effect that represents correlated residuals based on the distance between plots along both the rows and columns, where, \(\xi\sim N(0,\mathbf{R})\) and R is the covariance matrix of \(\xi\). Here, \(R ={\sigma_\xi^2\ \Sigma}_c\left(\rho_c\right)\otimes\Sigma_r\left(\rho_r\right)\). \(\sigma_\xi^2\) is the variance of spatially dependent residual; \({\Sigma}_c\left(\rho_c\right)\) and \(\ \Sigma_r\left(\rho_r\right)\) represents the first-order autoregressive correlation matrices and \(\rho_{c\ }\) and \(\rho_{r\ }\) are the autocorrelation parameters for the columns and rows; \(\otimes\) represents the Kronecker product between separable auto-regressive processes of the first order in the row-column dimensions.
R script in Asreml
model4<-asreml(fixed=trait~Block, random=~Genotype,residual =~ar1v(Block):ar1(Column), na.method=“include”, data=data)
- Block is fixed and genotype as random effects.
Model 5
- In this model we account for spatial trends i.e, correlated residuals across columns only.
\[ Y_{ijk}= G_{i} + B_{j}+ C_{k} + \xi\\ Y_{ijk}= \text{ is the effect of $i$th genotype and $j$th block in $k$th column} \\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ B_{k}= \text {fixed efect of $k$th block}\\ C_{k}= \text {efect of $k$th column}\\ \xi=\text {residual error}\\ \]
here, we assume \(\xi\) is a random effect that represents correlated residual across columns only, where, \(\xi\sim N(0,\mathbf{R})\) and R is the covariance matrix of \(\xi\), and \(\mathbf{R}={\sigma_\xi^2\ \Sigma}_c\left(\rho_c\right)\otimes I_r\). \(\sigma_\xi^2\) is the variance of spatially dependent residual; \({\Sigma}_c\left(\rho_c\right)\) represents the first-order autoregressive correlation matrices and \(\rho_{c\ }\) the autocorrelation parameters for the columns only; \(I_r\) represents independently and identically distributed variance structure for rows.
R script in Asreml
model5<-asreml(fixed=trait~Block, random=~Genotype,residual =~idv(Block):ar1(Column), na.method=“include”, data=data))
- Block as fixed effect and genotype as random effects.
4.1.1 Best Model for Grain Yield
- Here we will first built a function to run all the five models.
- These five models will be run separately on environment 1 and environment 2 data.
- Best model will be selected based on AIC and residual plot information, and later used to extract BLUPs in two environments.
- Please click on the code button on right-side to see the model with lowest AIC value.
Read data filtered for outliers and built function for running models
Best Model for Grain Yield Under Environment 1
> # Now run above function to test various models for both environments and traits
> # For grain yield under environment 1
> output.env1.gy<-my.asreml(demo.env1, trait = "Yield")
Online License checked out Wed Mar 24 21:16:30 2021
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:30 2021
LogLik Sigma2 DF wall cpu
1 -2756.464 743920.6 376 21:16:30 0.0
2 -2748.409 609062.8 376 21:16:30 0.0
3 -2737.023 384902.4 376 21:16:30 0.0
4 -2722.847 144448.7 376 21:16:30 0.0
5 -2718.047 66552.3 376 21:16:30 0.0
6 -2718.015 62713.2 376 21:16:30 0.0
7 -2718.015 62655.8 376 21:16:30 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:32 2021
LogLik Sigma2 DF wall cpu
1 -2776.060 690216.5 379 21:16:32 0.0 (1 restrained)
2 -2765.985 573570.7 379 21:16:32 0.0
3 -2753.990 362051.8 379 21:16:32 0.0
4 -2739.469 131220.2 379 21:16:32 0.0
5 -2732.670 55797.3 379 21:16:32 0.0 (2 restrained)
6 -2732.319 59030.0 379 21:16:32 0.0 (2 restrained)
7 -2732.310 60102.0 379 21:16:32 0.0 (2 restrained)
8 -2732.309 60081.9 379 21:16:32 0.0 (2 restrained)
9 -2732.309 60081.9 379 21:16:32 0.0 (1 restrained)
10 -2732.309 60081.9 379 21:16:32 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:33 2021
LogLik Sigma2 DF wall cpu
1 -2777.951 713886.2 379 21:16:33 0.0 (1 restrained)
2 -2766.215 584008.2 379 21:16:33 0.0
3 -2753.507 366319.0 379 21:16:33 0.0 (1 restrained)
4 -2741.336 168486.5 379 21:16:33 0.0 (2 restrained)
5 -2732.910 73123.0 379 21:16:33 0.0 (2 restrained)
6 -2730.524 59868.9 379 21:16:33 0.0 (2 restrained)
7 -2730.215 62109.7 379 21:16:33 0.0 (1 restrained)
8 -2730.201 63635.3 379 21:16:33 0.0 (1 restrained)
9 -2730.171 64413.5 379 21:16:33 0.0
10 -2730.169 64416.5 379 21:16:33 0.0 (1 restrained)
11 -2730.160 64769.9 379 21:16:33 0.0
12 -2730.159 64759.5 379 21:16:33 0.0 (1 restrained)
13 -2730.156 64908.0 379 21:16:33 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:34 2021
LogLik Sigma2 DF wall cpu
1 -2756.839 757812.6 376 21:16:34 0.0
2 -2748.188 620990.1 376 21:16:34 0.0
3 -2736.723 400946.6 376 21:16:34 0.0
4 -2723.405 165883.1 376 21:16:34 0.0 (1 restrained)
5 -2717.533 86647.5 376 21:16:34 0.0 (1 restrained)
6 -2715.805 64432.4 376 21:16:34 0.0
7 -2715.725 69904.3 376 21:16:34 0.0
8 -2715.607 69894.4 376 21:16:34 0.0
9 -2715.596 71087.8 376 21:16:34 0.0
10 -2715.589 71117.9 376 21:16:34 0.0
11 -2715.587 71269.3 376 21:16:34 0.0
12 -2715.586 71242.1 376 21:16:34 0.0
13 -2715.586 71281.5 376 21:16:34 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:36 2021
LogLik Sigma2 DF wall cpu
1 -2755.285 746909.5 376 21:16:36 0.0
2 -2747.278 613923.8 376 21:16:36 0.0
3 -2736.080 393579.7 376 21:16:36 0.0
4 -2722.492 157334.6 376 21:16:36 0.0
5 -2717.491 75123.5 376 21:16:36 0.0
6 -2717.335 66304.9 376 21:16:36 0.0
7 -2717.333 65792.7 376 21:16:36 0.0
> # Extract the name of model that has lower AIC
> best.model.env1.gy<-colnames(output.env1.gy)[apply(output.env1.gy,1,which.min)]
> best.model.env1.gy
[1] "model1"Click on code icon on right side to see which model is best
Best Model for Grain Yield Under Environment 2
> # For grain yield under environment 2
> output.env2.gy<-my.asreml(demo.env2, trait = "Yield")
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:40 2021
LogLik Sigma2 DF wall cpu
1 -2728.248 691772.4 374 21:16:40 0.0
2 -2722.699 574840.4 374 21:16:40 0.0
3 -2713.184 357862.3 374 21:16:40 0.0
4 -2697.903 114819.9 374 21:16:40 0.0
5 -2693.571 54154.3 374 21:16:40 0.0
6 -2693.563 52509.2 374 21:16:40 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:41 2021
LogLik Sigma2 DF wall cpu
1 -2748.855 645634.8 377 21:16:41 0.0
2 -2741.746 546605.1 377 21:16:41 0.0
3 -2731.373 344533.1 377 21:16:41 0.0
4 -2715.475 103539.8 377 21:16:41 0.0
5 -2709.982 40203.5 377 21:16:41 0.0
6 -2709.815 38285.0 377 21:16:41 0.0
7 -2709.810 38502.0 377 21:16:41 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:42 2021
LogLik Sigma2 DF wall cpu
1 -2747.192 655268.4 377 21:16:42 0.0
2 -2738.579 543567.0 377 21:16:42 0.0
3 -2726.797 339750.4 377 21:16:42 0.0 (1 restrained)
4 -2710.762 138196.2 377 21:16:42 0.0 (1 restrained)
5 -2708.409 75708.1 377 21:16:42 0.0 (1 restrained)
6 -2706.031 49119.0 377 21:16:42 0.0 (1 restrained)
7 -2705.050 66933.3 377 21:16:42 0.0 (1 restrained)
8 -2704.969 70833.9 377 21:16:42 0.0 (1 restrained)
9 -2704.957 72307.1 377 21:16:42 0.0
10 -2704.956 72709.5 377 21:16:42 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:43 2021
LogLik Sigma2 DF wall cpu
1 -2726.128 695306.5 374 21:16:43 0.0
2 -2719.381 574462.1 374 21:16:43 0.0
3 -2708.377 357139.5 374 21:16:43 0.0 (1 restrained)
4 -2692.387 140077.6 374 21:16:43 0.0 (1 restrained)
5 -2690.407 76752.3 374 21:16:43 0.0
6 -2688.296 54569.5 374 21:16:43 0.0
7 -2687.674 70417.1 374 21:16:43 0.0
8 -2687.632 74106.1 374 21:16:43 0.0
9 -2687.626 75581.3 374 21:16:43 0.0
10 -2687.626 75939.8 374 21:16:43 0.0
Model fitted using the gamma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:44 2021
LogLik Sigma2 DF wall cpu
1 -2725.104 687241.7 374 21:16:44 0.0
2 -2718.935 571044.5 374 21:16:44 0.0
3 -2708.342 356549.1 374 21:16:44 0.0 (1 restrained)
4 -2692.660 141531.9 374 21:16:44 0.0 (1 restrained)
5 -2690.215 78139.6 374 21:16:44 0.0
6 -2688.383 60226.1 374 21:16:44 0.0
7 -2688.064 67226.9 374 21:16:44 0.0
8 -2687.877 77539.6 374 21:16:44 0.0
9 -2687.868 79880.3 374 21:16:44 0.0
10 -2687.867 80421.8 374 21:16:44 0.0
> # Extract the name of model that has lower AIC
> best.model.env2.gy<-colnames(output.env2.gy)[apply(output.env2.gy,1,which.min)]
> best.model.env2.gy
[1] "model5"Click on code icon on right side to see which model is best
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4.1.2 Extract BLUPs and Heritabilities
Here in this section we will select and run the best model and extract BLUPs (know more on BLUPs or BLUEs here).
Best model will be selected based on lower AIC values and also residual plot. Lower the AIC value better is the model. For example for grain yield under environment 1 Model 1 has lower AIC values, however, we will pick Model 4 as best model because its AIC values is very close to Model 1 but residual plot lookes much better than Model 1**
We will also calculate the heritability’s. Note we are dealing with trials that is un-replicated and has missing data, so we cannot use basic formula as: \(h{^2}= \frac{\sigma^{2}g}{\sigma^{2}g+\sigma^{2}e}\) to calculate heritability. Plus when we are dealing with spatial models or complex models, calculating heritability with this method is not appropriate.
Alternative method as described by Piepho and M€ohring (2007) is more appropriate for complex residual structures and unbalanced experimental designs. The equation is: \(H_{C}=1-\frac{\overline{V}_{BLUP}}{2\sigma^{2}g}\). Where \(\overline{V}_{BLUP}\) is mean variance difference of difference of two BLUP and \(\sigma^{2}g\) is variance of genotypes. Note this definition of heritability is related to reliability of breeding value predictions. For more details please check the Book: Genetic Data Analysis for Plant and Animal Breeding; Chapter 7 and this beautiful resource Summary of heritability equations
So in this section a developed function called my.blup which will be used to extract BLUPs and then heritability will be calculated by method described above.
> # Now select the best model to extract BLUPs for each trait and environment
> # First we will build again a function to extract BLUPs and heritability from best model
> my.blup<-function(model, data){
+ #p<-plot(varioGram(model))
+ # Now use predict function to return the list of three containing predicted values, and average S.E differnces
+ predicted.values<-predict(model, "Genotype", sed=T)
+ # Extract the BLUPs from above
+ blups<-predicted.values$pvals
+ # Now let us add the line designation names
+ # BLUPs with line names
+ #blups<-merge(data[,c(7,8,13,14)],blups, by="Genotype")
+ #blups<-blups[!duplicated(blups$Genotype), ]
+ # Calculate the heritability
+ # Simply based on the variance components
+ #heritability<-vpredict(model5, hA ~ V1/(V1 + V2+V3+V4+V5))
+ #H2<-heritability[1,1]*100
+ #the Reliazied heritability that is appropriate for complex residual structures and unbalanced experimental designs introduced by Cullis et al. (2006) and discussed by Piepho and M€ohring (2007):
+ # page 235
+ # First let us extract the vBLUp difference
+ avgsd<-predicted.values$avsed[2]
+ h2<- (1-((predicted.values$avsed[2])^2/((summary(model)$varcomp[1,1])*2)))*100
+ return(list(Heritability=h2, BLUPs=blups))
+ }
> # Now for grain yield under environment 1
> # Fitting Model 4 as its best.
> model4.gy.env1<-asreml(fixed=Yield~Block, random=~Genotype,
+ residual =~ar1v(Block):ar1(Column), na.method="include", data=demo.env1)
> # BLUPs and heritability for grain yield under Environment 1
> out.gy.env1<-my.blup(model4.gy.env1, demo.env1)
> out.gy.env1$Heritability
> blups.env1.gy<-out.gy.env1$BLUPs
> names(blups.env1.gy)[c(2,3)]<-c("blups.gy", "std.er.gy")
>
> # Now for grain yield under environment 2
> model5.gy.env2<-asreml(fixed=Yield~Block, random=~Genotype,
+ residual =~idv(Block):ar1(Column), na.method="include", data=demo.env2)
> # BLUPs and heritability for grain yield under drought
> out.gy.env2<-my.blup(model5.gy.env2, demo.env2)
> out.gy.env2$Heritability
> blups.env2.gy<-out.gy.env2$BLUPs
> # rename the columns and select appropriate columns
> names(blups.env2.gy)[c(2,3)]<-c("blups.gy", "std.er.gy")
> # Now let us combine all the BLUPs dataframes into one and save
>
> # Let us add environment information column to the extracted BLUEs data file
> blups.env1<-data.frame(cbind(data.frame(Environment=c(rep("Env1",nrow(blups.env1.gy)))), blups.env1.gy))
> # Now add line.type information
> blups.env1<-merge(demo.env1[,c(4,9)],blups.env1, by="Genotype")
> blups.env1<-blups.env1[!duplicated(blups.env1$Genotype), ]
> # Now combine environment 2 information
> blups.env2<-data.frame(cbind(data.frame(Environment=c(rep("Env2",nrow(blups.env2.gy)))), blups.env2.gy))
> # Now add the genotype names name and line.type
> blups.env2<-merge(demo.env2[,c(4,9)],blups.env2, by="Genotype")
> blups.env2<-blups.env2[!duplicated(blups.env2$Genotype), ]
> # Now combine all
> blups.all<-rbind(blups.env1[,-6], blups.env2[,-6])
> # Round all the columns containing blups and standard errors
> blups.all<-data.frame(lapply(blups.all, function(y) if(is.numeric(y)) round(y, 2) else y))
> # Save the blups in the directory
> write.csv(blups.all,
+ file="~/Documents/GitHub/Analysis-pipeline/Outputs/Tables/blups.all.seperate.csv",
+ row.names = FALSE)Summary and Heritability for Grain Yield
> # Calcualate summary and heritability
> # Save heritability as vector
> Heritability<-c(out.gy.env1$Heritability, out.gy.env2$Heritability)
> #
> summary.gy<-cbind(data.frame(blups.all%>%
+ group_by(Environment)%>%
+ summarize(Mean = mean(blups.gy, na.rm=TRUE),
+ Median= median(blups.gy, na.rm=TRUE),
+ SD =sd(blups.gy, na.rm=TRUE),
+ Min.=min(blups.gy, na.rm=TRUE),
+ Max.=max(blups.gy, na.rm=TRUE))
+ ),Heritability)
> # Round
> summary.gy<-data.frame(lapply(summary.gy, function(y) if(is.numeric(y)) round(y, 2) else y))
> # Plot the data.tables
> print_table(summary.gy, rownames = FALSE,caption = htmltools::tags$caption("Data summaries of BLUPs and heritability for grain yield (kg/ha) in two environments.", style="color:black; font-size:130%"))Note: Heritability under two environments is very close.
BLUPs for Grain Yield
4.2 Multi-environment trial (MET)/Combined Analysis
Here in this section MET analysis for two environments will be performed and single value BLUP for each genotype will be predicted. We will also calculate combined heritability.
With the separate analysis done above we know which best spatial model works in each trial. We will directly borrow this information and incorporate into our combined model analysis, so we do not need to test various models.
*Click button below for more model description:
Combined or MET model
\[ Y_{ijk}= \mu+G_{i} + E_{j}+G_{i}*E_{j}+ B_{k}(E_{j}) + \xi\\ Y_{ijk}= \text{ is the effect of $i$th genotype in $j$th environment and $k$th block} \\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ E_{j}=\text{fixed effect of the $j$th environment}\\ B_{k}= \text {random efect of $k$th block nested in $j$th environment}\\ \xi=\text{residual error}\\ \]
here, we assume \(\xi\) is a random effect that represents correlated residual across columns and or rows depending upon the correlation structure that was fitted. Further it should be noted we applied a separate spatial trend for each environment in the combined model, and spatial model was decided from the seperate analysis done above. See the Asreml code below:
R script in Asreml for grain yield
met.gy<-asreml(Yield ~Environment,random= ~Genotype +Environment:Genotype+Block:Environment,residual = dsum(ar1v(Block):ar1(Column)+ ~idv(Block):ar1(Column)|Environment,levels = list(c(1), c(2))), na.method =“include”, data = crurrs.data.out)
- Genotype,block were treated as random effects and environments as fixed effect .
> # First we will read the filtered data set containing data from two environments.
> if(exists('demo.data.out') && is.data.frame(get('demo.data.out'))){
+ demo.data.out=demo.data.out
+ }else{
+ demo.data.out<-read.csv(file="~/Documents/GitHub/Analysis-pipeline/Outputs/Tables/demo.data.filtered.csv",
+ header = TRUE)
+ }
> # In case checks are used as fixed effects
> # Create two new columns if design is augmented.
> # Adding a new column 'new' that will help treat genotypes as separate
> #demo.data.out$Genotype<-as.numeric(demo.data.out$Genotype)
> #demo.data.out<- within(demo.data.out,{
> #new <- ifelse(demo.data.out$Line.type=="check", 0, 1)
> #})
> # Adding a new column 'Genotypec' that will help us group all the new entries
> # in a single pool, yet treat all checks as separate
> #demo.data.out<- within(demo.data.out, {
> # Genotypec <- ifelse(demo.data.out$new > 0, 999, demo.data.out$Genotype)
> #})
> # Arrange the the data set before running it
> demo.data.out<-data.frame(demo.data.out%>% group_by(Environment)%>%arrange(Row, Column))
> demo.data.out<-demo.data.out%>% arrange(Environment)
> columns<-c("Plot", "Genotype", "Replication", "Block", "Row", "Column", "Year")
> demo.data.out[, columns]<-lapply(columns, function(x) as.factor( demo.data.out[[x]]))
> demo.data.out$Yield<-as.numeric(demo.data.out$Yield)
> demo.data.out$Environment<-as.factor(demo.data.out$Environment)Click on code on right-side to see detailed models and how heritability and BLUPs are extracted
> # Here we will perform combined analysis of two environmenys
> # Spatial variation model will be used, model will be selected based on previous analysis done seperately
> # For GRAIN YIELD
> met.gy<-asreml(Yield ~1,random= ~Genotype +Environment:Genotype+Block:Environment,
+ residual =~dsum(~ar1v(Block):ar1(Column)+idv(Block):ar1(Column)|Environment,levels = list(c(1), c(2))), na.method ="include", data = demo.data.out)
Multi-section model using the sigma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:48 2021
LogLik Sigma2 DF wall cpu
1 -5984.741 1.0 757 21:16:49 0.1
2 -5770.608 1.0 757 21:16:49 0.1
3 -5542.038 1.0 757 21:16:49 0.1 (1 restrained)
4 -5414.267 1.0 757 21:16:49 0.1
5 -5332.976 1.0 757 21:16:49 0.1
6 -5298.165 1.0 757 21:16:49 0.1
7 -5294.841 1.0 757 21:16:49 0.1
8 -5293.464 1.0 757 21:16:49 0.1
9 -5293.219 1.0 757 21:16:49 0.1
10 -5293.204 1.0 757 21:16:49 0.1
11 -5293.200 1.0 757 21:16:49 0.1
12 -5293.199 1.0 757 21:16:49 0.1
13 -5293.199 1.0 757 21:16:50 0.1
>
> #aic<- -2*(model.met2$loglik-length(model.met2$vparameters));aic
> predicted.gy<-predict(met.gy, "Genotype", sed=T)
Multi-section model using the sigma parameterization.
ASReml 4.1.0 Wed Mar 24 21:16:50 2021
LogLik Sigma2 DF wall cpu
1 -5293.199 1.0 757 21:16:50 0.1
2 -5293.199 1.0 757 21:16:50 0.1
3 -5293.199 1.0 757 21:16:50 0.2
> # Extract the BLUPs from above
> blups.gy.met<-predicted.gy$pvals
> names(blups.gy.met)[c(2,3)]<-c("blups.gy", "std.er.gy")
> # Now calculate heritability
> h2.gy.met<- (1-((predicted.gy$avsed[2])^2/((summary(met.gy)$varcomp[2,1])*2)))*100;h2.gy.met
mean
87.10002
> # Now add designation and line.type to blup file
> # Now add the genotype name and line.type
> blups.met<-merge(demo.env1[,c(4,9)],blups.gy.met[,-4], by="Genotype")
> blups.met<-blups.met[!duplicated(blups.met$Genotype), ]
> blups.met<-data.frame(lapply(blups.met, function(y) if(is.numeric(y)) round(y, 2) else y))BLUPs for grain yield from combined analysis
> # BLUPs table
> print_table(blups.met[, c(1,2, 3,4)],editable = 'cell', rownames = FALSE,caption = htmltools::tags$caption(" BLUPs along with standard errors for grain yield (kg/ha) from MET analysis", style="color:black; font-size:130%"))> # Save the blup file
> write.csv(blups.met,
+ file="~/Documents/GitHub/Analysis-pipeline/Outputs/Tables/blups.combined.csv",
+ row.names = FALSE)Combined Data Summary and Heritability
> summary.met.gy<-data.frame(blups.met%>%
+ group_by(Line.type)%>%
+ summarize(Mean = mean(blups.gy, na.rm=TRUE),
+ Median= median(blups.gy, na.rm=TRUE),
+ SD =sd(blups.gy, na.rm=TRUE),
+ Min.=min(blups.gy, na.rm=TRUE),
+ Max.=max(blups.gy, na.rm=TRUE),
+ Heritability=h2.gy.met)
+ )
> summary.met.gy<-data.frame(lapply(summary.met.gy, function(y) if(is.numeric(y)) round(y, 2) else y))
> summary.met.gy[1,7]<-"-"
> print_table(summary.met.gy, rownames = FALSE)Note: Heritability of grain yield in combined analysis is low as compared to seperate analysis. Also, check the mean and higher value of genotype as compared to checks.
5 Phenotypic Data Analysis in lme4 R Package
Here in this section phenotypic data analysis is performed in an open source R package called lme4. More on this R package can be found here lme4 Tutorial 1, and lme4 Tutorial 2.
The purpose of this section is to repeat the phenotypic data analysis in lme4 as ASReml R package is commercial package and may not available for all the users.
Filtered data set will be used, same one used in ASReml R package to perform the analysis in lme4.
ANOVA, variance components, BLUPS, BLUES and heritability is extracted for the results part.
5.1 Upload the Filtered Phenotypic Data
> # Read the saved csv file, if working directly
> if(exists('demo.data.out') && is.data.frame(get('demo.data.out'))){
+ demo.data.out=demo.data.out
+ }else{
+ demo.data.out<-read.csv(file="~/Documents/GitHub/Analysis-pipeline/Outputs/Tables/demo.data.filtered.csv",
+ header = TRUE)
+ # factor conversion if below are not in factors
+ columns<-c("Plot", "Genotype", "Replication", "Block", "Row", "Column", "Year")
+ demo.data.out[, columns]<-lapply(columns, function(x) as.factor(demo.data.out[[x]]))
+ demo.data.out$Yield<-as.numeric(demo.data.out$Yield)
+ demo.data.out$Height<-as.numeric(demo.data.out$Height)
+ }
> # Now we will subset the environments and traits for analysis
> demo.data.out<-data.frame(demo.data.out%>% group_by(Environment)%>%arrange(Row, Column)) # arrange by row and column
> demo.data.out<-data.frame(demo.data.out%>% arrange(Environment)) # Arrange by environment
> demo.env1<-subset(demo.data.out, Environment=="Env1") # Environment 1
> demo.env1<-droplevels.data.frame(demo.env1)
> demo.env2<-subset(demo.data.out, Environment=="Env2") # Environment 2
> demo.env2<-droplevels.data.frame(demo.env2)5.2 Seperate Analayis
Model 1.lme4
\[ Y_{ij}= \mu+G_{i} + B_{j} + \varepsilon_{ij}\\ Y_{ij}= \text{ is the effect of $i$th genotype in $j$th block}\\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ B_{j}= \text {fixed effect of $j$th block}\\ e_{ij}=\text{error}\\ \text{here we assume residuals are independent and identically distributed }\varepsilon\sim \text{$iid$N}(0,\sigma_e^2)\\ \]
- The model described above is equivalent to model1 described in ASReml R package analysis.
5.2.1 Results
- Here we will summarize the results using summary() function. The first few lines of output indicate that the model was fitted by REML as well as the value of the REML criterion. The second piece of the summary output provides information regarding the random-effects and residual variation. The third piece of the summary output provides information regarding the fixed-effects and the fourth piece of summary output provides information regarding the correlation of fixed effects.
> # Summarise the results
> summary(model1.lme4)
Linear mixed model fit by REML ['lmerMod']
Formula: Yield ~ Block + (1 | Genotype)
Data: demo.env1
REML criterion at convergence: 6127.1
Scaled residuals:
Min 1Q Median 3Q Max
-1.77102 -0.20285 0.00171 0.19963 1.82542
Random effects:
Groups Name Variance Std.Dev.
Genotype (Intercept) 747281 864.5
Residual 62655 250.3
Number of obs: 380, groups: Genotype, 344
Fixed effects:
Estimate Std. Error t value
(Intercept) 4061.23 70.00 58.015
Block2 -100.16 82.48 -1.214
Block3 14.68 82.48 0.178
Block4 21.22 82.48 0.257
Correlation of Fixed Effects:
(Intr) Block2 Block3
Block2 -0.589
Block3 -0.589 0.500
Block4 -0.589 0.500 0.5005.2.1.1 Extract variance components
5.2.1.2 Plot the residual vs fitted plot (check for homoscedasticicty)
> # Residual plot
> plot(residuals(model1.lme4,type="pearson"), main='Model residuals',
+ ylab='Pearson residual value')
5.2.1.3 ANOVA
5.2.1.4 Extract the Fixed effects
5.2.2 Heritability
> # Extract the variance components
> Ve<- data.frame (VarCorr(model1.lme4))
> Ve
grp var1 var2 vcov sdcor
1 Genotype (Intercept) <NA> 747281.11 864.4542
2 Residual <NA> <NA> 62655.26 250.3103
> # Now calculate heritability using variance components
> genotype.var=Ve[1,4]
> error.var=Ve[2,4]
> # Now heritability
> h2=genotype.var/(genotype.var+error.var)*100
> h2
[1] 92.26417
> # Reliability
> std.err<-se.ranef(model1.lme4)$Genotype
> v_BLUP<- mean(std.err)
> # Heritability/Reliability
> h2<- (1-((v_BLUP)^2/(Ve[1,4]*2)))*100
> h2
[1] 96.261815.3 Stage Wise/Combined Analysis
Model 2.lme4
- Here we will analyze the two environments and extract the single BLUPs for each genotype. We will use mixed model analysis in lme4 r package model. We will treat genotypes as random.
5.3.1 Combined ANOVA
> demo.data.out$Environment<-as.factor(demo.data.out$Environment)
> model.anova<-lm(formula = Yield~Genotype+Environment+Genotype:Environment+Environment/Block,data=demo.data.out)
> # Get ANOVA
> anova(model.anova)
Analysis of Variance Table
Response: Yield
Df Sum Sq Mean Sq F value Pr(>F)
Genotype 343 554389253 1616295 28.2725 < 2.2e-16 ***
Environment 1 1049221 1049221 18.3532 6.092e-05 ***
Genotype:Environment 341 67400900 197657 3.4574 1.023e-08 ***
Environment:Block 6 505160 84193 1.4727 0.2012
Residuals 66 3773115 57168
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 15.3.2 Check for Homogeneity of Variance
> var.test(Yield~Environment,data=demo.data.out)
F test to compare two variances
data: Yield by Environment
F = 1.0719, num df = 379, denom df = 377, p-value = 0.4998
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
0.875926 1.311767
sample estimates:
ratio of variances
1.071949 5.3.3 Combined Analysis in lme4
- The model we will use is give below:
\[ Y_{ij}= \mu + G_{i} + E_{j} + G_{i}*E_{j}+ B_{k}(E_{j})+ \varepsilon_{ijk}\\ Y_{ij}= \text{ is the effect of $i$th genotype in $j$th environment and $k$th block} \\ \mu= \text {overall mean}\\ G_{i}=\text{random effect of the $i$th genotype}\\ E_{j}= \text {fixed effect of $j$th environment}\\ G_{i}*G_{j}= \text {interaction effect of $i$th genotype in $j$th environment}\\ B_{k}(E_{j})= \text {effect of $k$th block nested with $j$th environment}\\ e_{ijk}=\text{error}\\ \text{here we assume residuals are independent and identically distributed }\varepsilon\sim \text{$iid$N}(0,\sigma_e^2)\\ \]
Mixed models are powerful tools to handle assumptions of linear model Read this one
We will extract variance components and also calculate heritability.
> demo.data.out$Environment<-as.factor(demo.data.out$Environment)
> Model2.lme4<-lmer(Yield ~Environment/Block+(1|Genotype)+(1|Environment)+(1|Genotype:Environment), data=demo.data.out)
> summary(Model2.lme4)
Linear mixed model fit by REML ['lmerMod']
Formula: Yield ~ Environment/Block + (1 | Genotype) + (1 | Environment) +
(1 | Genotype:Environment)
Data: demo.data.out
REML criterion at convergence: 11898.4
Scaled residuals:
Min 1Q Median 3Q Max
-2.00119 -0.26573 -0.01489 0.28043 1.80033
Random effects:
Groups Name Variance Std.Dev.
Genotype:Environment (Intercept) 125837 354.7
Genotype (Intercept) 619296 787.0
Environment (Intercept) 63883 252.8
Residual 56732 238.2
Number of obs: 758, groups:
Genotype:Environment, 686; Genotype, 344; Environment, 2
Fixed effects:
Estimate Std. Error t value
(Intercept) 4041.767 260.412 15.521
EnvironmentEnv2 -147.060 363.246 -0.405
EnvironmentEnv1:Block2 -33.050 65.447 -0.505
EnvironmentEnv2:Block2 60.946 65.541 0.930
EnvironmentEnv1:Block3 3.333 65.448 0.051
EnvironmentEnv2:Block3 115.206 65.613 1.756
EnvironmentEnv1:Block4 48.956 65.445 0.748
EnvironmentEnv2:Block4 234.575 65.559 3.578
Correlation of Fixed Effects:
(Intr) EnvrE2 EE1:B2 EE2:B2 EE1:B3 EE2:B3 EE1:B4
EnvrnmntEn2 -0.697
EnvrnmE1:B2 -0.126 0.089
EnvrnmE2:B2 -0.004 -0.088 0.008
EnvrnmE1:B3 -0.126 0.085 0.499 0.041
EnvrnmE2:B3 -0.004 -0.088 0.018 0.501 0.028
EnvrnmE1:B4 -0.126 0.088 0.499 0.015 0.499 0.015
EnvrnmE2:B4 -0.003 -0.088 0.003 0.503 0.043 0.501 0.008
optimizer (nloptwrap) convergence code: 0 (OK)
unable to evaluate scaled gradient
Model failed to converge: degenerate Hessian with 1 negative eigenvalues
> plot(Model2.lme4)
> # Extract the variance components
> Ve<- data.frame (VarCorr(Model2.lme4))
> Ve
grp var1 var2 vcov sdcor
1 Genotype:Environment (Intercept) <NA> 125837.07 354.7352
2 Genotype (Intercept) <NA> 619296.17 786.9537
3 Environment (Intercept) <NA> 63883.47 252.7518
4 Residual <NA> <NA> 56732.25 238.1853
> # Heritability
> std.err<-se.ranef(Model2.lme4)$Genotype
> v_BLUP<- mean(std.err)
> # Heritability/Reliability
> h2<- (1-((v_BLUP)^2/(Ve[2,4]*2)))*100
> h2
[1] 93.482386 Ranking of Genotypes and Correlations
Here in this section we will rank the genotypes based on the BLUPs extracted from the combined analysis in ASReml R package and select top 10% of genotypes and plot it as bar plot. Same thing can be done with BLUPs extracted from the lme4 R package.
We will also compare the rankings of genotypes based on BLUPs obtained in environment 1, environment 2 and combined analysis. We will see which genotypes are common in top 10% of lines all the three. We will save it in data.frame and also plot Venn Diagram.
We will also check the correlations between BLUPs in Env1, Env2 and combined one.
6.1 Ranking of BLUPs
- Ranking of genotypes based the BLUPs extracted in Combined analysis
> # Ranking and selection of top performing lines
> # Subset only entries
> blups.met.Genotype<-subset(blups.met, Line.type=="entry")
> # Get mean of entries and checks
> Genotype.mean<-mean(blups.met.Genotype$blups.gy)
> check.mean<-mean((subset(blups.met, Line.type=="check"))$blups.gy)
> # Arrange the BLUPs in decreasing order
> blups.met.Genotype<-blups.met.Genotype%>%arrange(desc(blups.gy))
> # Select top 35 and merge with checks
> blups.top25<-data.frame(rbind((blups.met.Genotype[1:35, ]), (subset(blups.met, Line.type=="check"))))
> blups.top25<-droplevels.data.frame(blups.top25)
> # make factor unique to keep order of entries on plot
> blups.top25$Genotype <- factor(blups.top25$Genotype, levels=unique(blups.top25$Genotype))
> # Draw the plot
> bar.plot<-ggplot(data=blups.top25, aes(x=Genotype, y=blups.gy, fill=Line.type)) +
+ geom_bar(stat="identity", width=0.5)+
+ theme_classic()+
+ labs(title="BLUPs of Top Ranked Genotypes along with Checks",x="Genotypes", y = "BLUP value")+
+ #scale_y_continuous(limits = c(0, 6000), breaks = seq(0, 6000, by = 500))+
+ theme (plot.title = element_text(color="black", size=1, face="bold", hjust=0),
+ axis.title.x = element_text(color="black", size=10, face="bold"),
+ axis.title.y = element_text(color="black", size=10, face="bold")) +
+ theme(axis.text= element_text(color = "black", size = 8))+
+ geom_segment(aes(x = 1, y = Genotype.mean, xend = 35, yend =Genotype.mean), color="darkred",
+ linetype="dashed", size=1)+
+ geom_segment(aes(x = 36, y = check.mean, xend = 47, yend =check.mean), color="darkblue",
+ linetype="dashed", size=1)+
+ theme(axis.text.x = element_text(angle = 90, hjust = 1))
> ggplotly(bar.plot)Bar plot showing the BLUPs for top 10% of genotypes and all the checks. Dashed lines shows overall mean of all genotypes and checks. Genotypes differ slightly from checks and mean of entries and checks are almost close.
6.2 Correlation Between BLUPs
- Here checking the correlation of BLUPs obtained in Environment 1, Environment 2 and Combined analysis
> # BLUPs in Environment 1
> blups.env1<-subset(blups.all, Environment=="Env1", select =c(1,4))
> colnames(blups.env1)<-c("Genotype", "BLUPs.Env1")
> # Blups in Environment 2
> blups.env2<-subset(blups.all, Environment=="Env2", select =c(1,4))
> colnames(blups.env2)<-c("Genotype", "BLUPs.Env2")
> # Now combined blups
> blups.com<-blups.met[, c(1,2,3)]
> colnames(blups.com)<-c("Genotype", "Line.type", "BLUPs.combined")
> # Merge all the BLUPs
> blups.com.all<-merge((merge(blups.env1, blups.env2, by="Genotype")), (blups.com), by="Genotype")
> str(blups.com.all)
'data.frame': 344 obs. of 5 variables:
$ Genotype : Factor w/ 344 levels "1","2","3","4",..: 1 10 100 101 102 103 104 105 106 107 ...
$ BLUPs.Env1 : num 3303 3757 5260 2725 5273 ...
$ BLUPs.Env2 : num 3010 3499 4018 3414 3977 ...
$ Line.type : chr "entry" "entry" "entry" "entry" ...
$ BLUPs.combined: num 3163 3559 4676 3047 4563 ...
> corr.blup <- data.frame(round(cor(blups.com.all[,-c(1,4)]), 2))
> print_table(corr.blup, rownames = TRUE, caption = htmltools::tags$caption("Correlation of BLUPs obtained in seperate analysis for environment 1, environment 2 and in combined analysis.", style="color:black; font-size:130%"))6.3 Venn Diagram of Top Lines
- Here we will look at the lines that are common in top 10% of lines in Environment 1, Environment 2 and combined analysis.
> # Combined blups
> com.blups.top<-subset(blups.met, Line.type=="entry")
> com.blups.top<-com.blups.top%>%arrange(desc(blups.gy))
> com.blups.top<-com.blups.top[1:35,]
> colnames(com.blups.top)[1]<-"Genotype.com"
>
> # Blups in Environment 1
> blups.env1<-subset(blups.all, Environment=="Env1", select =c(1,4))
> blups.env1.top<-blups.env1%>%arrange(desc(blups.gy))
> blups.env1.top<-blups.env1.top[1:35,]
> colnames(blups.env1.top)[1]<-"Genotype.Env1"
> # Blups in Environment 2
> blups.env2<-subset(blups.all, Environment=="Env2", select =c(1,4))
> blups.env2.top<-blups.env2%>%arrange(desc(blups.gy))
> blups.env2.top<-blups.env2.top[1:35,]
> colnames(blups.env2.top)[1]<-"Genotype.Env2"
> # Now cbinb all the required columns
> data.venn<-data.frame(cbind(Combined=com.blups.top$Genotype.com, Environment1=blups.env1.top$Genotype.Env1, Environment2=blups.env2.top$Genotype.Env2))
> myCol <- brewer.pal(3, "Pastel2")
> P<-venn.diagram(
+ x = list(data.venn$Combined, data.venn$Environment1, data.venn$Environment2),
+ category.names = c("Combined.BLUPs" , "Env1.BLUPs " , "Env2.BLUPs"),
+ filename = '~/Documents/GitHub/Analysis-pipeline/Codes/14_venn_diagramm.png',
+ output=TRUE,
+ # Output features
+ imagetype="png" ,
+ height = 1200 ,
+ width = 1200 ,
+ resolution = 500,
+ # Circles
+ lwd = 2,
+ lty = 'blank',
+ fill = myCol,
+
+ # Numbers
+ cex = .8,
+ fontface = "bold",
+ fontfamily = "sans",
+
+ # Set names
+ cat.cex = 0.4,
+ cat.fontface = "bold",
+ cat.default.pos = "outer",
+ cat.pos = c(-27, 27, 135),
+ cat.dist = c(0.055, 0.055, 0.085),
+ cat.fontfamily = "sans",
+ rotation = 1
+
+ )
> P
[1] 1
Venn diagram showing list of lines that are common between separate analysis in environment 1, environment 2 and combined analysis. 17 top ranking genotypes were found common in environment 1, environment 2 and combined analysis and can be used for selection for broad adoptation
6.4 List of lines that are common in all the three
> overlap <- calculate.overlap(
+ x = list(data.venn$Combined, data.venn$Environment1, data.venn$Environment2)
+ )
> datatable(t(overlap$a5), rownames = TRUE, caption = htmltools::tags$caption("List of entries that are common between Environment 1, Environment 2, and in combined data analysis", style="color:black; font-size:130%"))Table showing list of entries (genotype number is shown) that are common in all the three.
7 Final Remarks
Models accounting for spatial trends were more appropriate than models accounting just for experimental design factors.
Combined analysis performed would be more appropriate to rank lines, as done above.
8 Literature
Analysis and Handling of G × E in a Practical Breeding Program
A stage‐wise approach for the analysis of multi‐environment trials
Experimental design matters for statistical analysis: how to handle blocking
Random effects structure for confirmatory hypothesis testing: Keep it maximal
Generalized linear mixed models: a practical guide for ecology and evolution
Perils and pitfalls of mixed-effects regression models in biology
A brief introduction to mixed effects modelling and multi-model inference in ecology
Modeling Spatially Correlated and Heteroscedastic Errors in Ethiopian Maize Trials
More, Larger, Simpler: How Comparable Are On‐Farm and On‐Station Trials for Cultivar Evaluation
Rethinking the Analysis of Non‐Normal Data in Plant and Soil Science
Fundamentals of Experimental Design: Guidelines for Designing Successful Experiments
Note: For questions specific to data analysiss shown here contact waseem.hussain@irri.org
If your experiment needs a statistician, you need a better experiment - Ernest Rutherford